PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 69, Number 2, May 1978 A NONSTANDARD CHARACTERIZATION OF WEAKCONVERGENCE ROBERT M. ANDERSON AND SALIM RASHID1 Abstract. Let A" be any topological space, and C(X) the space of bounded continuous functions on X. We give a nonstandard characterization of weak convergence of a net of bounded linear functionals on CiX) to a tight Baire measure on X. This characterization applies whether or not the net or the individual functionals in the net are tight. Moreover, the characterization is expressed in terms of the values of an associated net of countably additive measures on all Baire sets of X; no distinguished family, such as the family of continuity sets of the limit, is involved. As a corollary, we obtain a new proof that a tight set of measures is relatively weakly compact. 1. Introduction. Let X be an arbitrary topological space, CiX) the Banach space of all bounded continuous real-valued functions on X (with the sup norm), and CiX)d the space of bounded linear functionals on CiX). The weak topology (or more precisely, the weak-star topology) on CiX)d is the topology generated by the subbase {{</>: <p(/) < a),f E CiX), a E R). Thus, a net {<t>a}aeDconverges weakly to d> (written </>„=><>) if and only if $aif) ~*"K/) f°r all/ e CiX). This notion is of fundamental importance in probability, where one considers the weak topology relativized to the space of all Baire (or Borel) probability measures on X. If ( ju.a)is a net of measures, ¡xa =>ft if and only if ffd¡ia —> //d\i for all/ E CiX). In this paper, we give a nonstandard characterization of weak convergence of a net {</>a}a(EO in CiX)d to a tight countably additive Baire measure. The characterization involves an associated net {fia}ae*o OI" countably additive Baire measures on X. \ia is obtained from a Loeb measure as introduced by Peter A. Loeb [5] and a measure-preserving map. Measure-preserving maps in nonstandard analysis were first used in special cases by Loeb [6] followed by the first author [1]; a general theory based on them will be given in [2]. One aspect of the characterization is particularly noteworthy. Even in the Received by the editors May 18, 1977and, in revised form, September 21, 1977. AMS (A/OS)subject classifications (1970). Primary 02H25, 26A98, 60B05, 60B10. Key words and phrases. Weak convergence, tight, relatively weakly compact, topological measure theory, nonstandard analysis. 'Portions of this material were developed independently by the authors and included in dissertations submitted for the degree of Doctor of Philosophy in Yale University. The authors are grateful to Professors Yoav Benyamini, Donald J. Brown, S. Kakutani and Ward Whitt for a number of very profitable discussions, and to the referee for a number of suggestions which improved the exposition. The first author gratefully acknowledges the support of a Canada Council Doctoral Fellowship. © American Mathematical Society 1978 327 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use